The Density of the Set of Trisectable Angles

TL;DR

This paper investigates the density and distribution of angles that can or cannot be trisected with straightedge and compass, revealing that the set of trisectable angles is very sparse within the unit circle.

Contribution

It introduces the concept of computational density for trisectable angles within algebraic number fields and proves it is zero for fields of degree up to two.

Findings

Trisectable angles form a topologically dense but measure-zero subset of the circle.

The set of trisectable angles with algebraic cosine values is very thin and conjectured to have zero density.

Proves the zero density of trisectable angles in quadratic fields using elementary counting and number theory.

Abstract

It has been known for almost 200 years that some angles cannot be trisected by straightedge and compass alone. This paper studies the set of such angles as well as its complement $\mathcal{T}$, both regarded as subsets of the unit circle $S^1$. It is easy to show that both are topologically dense in $S^1$ and that $\mathcal{T}$ is contained in the countable set $\mathcal{A}$ of all angles whose cosines (or, equivalently, sines) are algebraic numbers (Corollary 3.2). Thus, $\mathcal{T}$ is a very "thin" subset of $S^1$. Pushing further in this direction, let $K$ be a real algebraic number field, and let $\mathcal{T}_K$ denote the set of trisectable angles with cosines in $K$. We conjecture that the "computational density" of $\mathcal{T}_K$ in $K$ is zero and prove this when $K$ has degree $\leq 2$ (cf. \S 1.2 and Theorem 4.1). In addition to some introductory field theory, the paper uses elementary counting arguments to generalize a theorem of Lehmer (Theorem 5.2)on the density of the set of relatively prime $n$-tuples of positive integers.